Area and Circumference of Circles

By the end of this lesson you will be able to:

• State and apply the circumference formula
• State and apply the area formula
• Explain informally where both formulas come from
• Choose the right formula for any real-world context

What You Will Learn Today

1. State the circumference formula and identify each variable
2. State the area formula and identify each variable
3. Compute circumference and area — exact and approximate forms
4. Explain informally why using the definition of π
5. Solve real-world problems involving circumference and area
6. Distinguish circumference (linear) from area (square) and choose correctly

Where Do You Already See Circles?

A bicycle wheel rolls forward. A pizza gets sliced. A clock marks the hours.

• Radius (): distance from center to edge
• Diameter (): distance across the circle —
• Today's question: how do we measure the distance around and the space inside?

What Happens When We Measure C ÷ d?

Wrap a string around a circle to measure . Measure with a ruler. Divide: .

What do you notice about the last column?

The Ratio That Never Changes — π

No matter the size of the circle, — this ratio is called π (pi)

The Circumference Formula Comes from π

• π ≈ 3.14159... — irrational; 3.14 is an approximation
• Exact: leave as π (e.g., cm)
• Approximate: substitute 3.14 (e.g., ≈ 43.96 cm)

Worked Example 1: Circumference from Diameter

Problem: A circle has diameter cm. Find the circumference.

• Exact: cm
• Approximate: cm
• Units: centimeters — circumference is a distance (not cm²)

Worked Example 2: Circumference from Radius

Problem: A circle has radius in. Find the circumference.

• Exact: in (Approximate: in)
• Radius given → . Diameter given → .

Worked Example 3: Finding Diameter from Circumference

Problem: A circular track has circumference m. Find the diameter.

• Strategy: divide by π to find diameter
• Check: m ✓

Check: Find the Exact Circumference

A circle has radius cm.

1. Which formula do you use?
2. Find the exact circumference (in terms of π)
3. Find the approximate circumference (use π ≈ 3.14)

Answer: cm

Practice: Five Circumference Calculation Problems

Find the circumference. Give exact and approximate answers (π ≈ 3.14).

1. cm
2. m
3. ft
4. in
5. Inverse: cm → find the radius

Answers: Check Your Circumference Practice

1. cm
2. m
3. ft
4. in
5. cm

Watch for: using diameter directly in C = 2πr, or forgetting units.

Bridging from Circumference to Area

We know the distance around a circle. Now: how do we find the space inside?

• Rectangles: area = length × width — easy
• Circles: curved edges — can't just multiply sides
• Big idea: cut a circle into sectors, rearrange — a familiar shape appears

Sector Rearrangement — Building a Rectangle

Flip alternating sectors to form a rough parallelogram — the more sectors, the closer to a rectangle

The Rectangle Dimensions Reveal the Formula

• Length ≈ half the circumference
• Width ≈ the radius

Use radius only — square the radius first, then multiply by π

Worked Example 1: Area from Radius

Problem: A circle has radius cm. Find the area.

• Exact: cm²
• Approximate: cm²
• cm² — area is a 2D region, always square units

Worked Example 2: Area from Diameter

Problem: Circular tabletop, ft. Find the area.

Step 1: ft

Given diameter: find first, then apply

Worked Example 3: Finding Radius from Area

Problem: A circle has area m². Find the radius.

Check: m² ✓

Strategy: divide by π first, then take the square root

Check: Finding Area from Diameter

A circle has diameter cm. Do not skip step 1!

1. What is the radius?
2. Find the exact area
3. Find the approximate area (use π ≈ 3.14)

Answer: ; cm²

Practice: Five Area Calculation Problems

Find the area. Give exact and approximate answers (π ≈ 3.14).

1. cm
2. m
3. ft
4. in
5. Inverse: m² → find the radius

Answers: Check Your Area Practice

1. cm²
2. m²
3. ; ft²
4. ; in²
5. ; m

Watch for: using diameter without halving (3–4), skipping the square root (5), writing cm not cm².

Two Formulas — When Do You Use Each?

• Circumference → linear units (cm, m, ft)
• Area → square units (cm², m², ft²)
• Key question: Am I measuring around the edge or filling the inside?

Formula Selector: Going Around or Covering Inside?

• Going around / border / perimeter →
• Covering / filling / inside →

Sorting Scenarios — Circumference or Area?

Classify each scenario without computing:

"around / border / distance" → C. "Fill / cover / material" → A.

Worked Example: Fencing a Circular Garden

Garden, ft — how much fencing?

"Fencing" → going around → circumference

Worked Example: Grass Seed for a Circular Lawn

Problem: Same garden, ft. How much grass seed is needed?

"Grass seed" = covering the interior → use area

You need seed for approximately 113.1 square feet.

Same circle, same radius — different formula because the question changed.

Check: One Bicycle Wheel Rotation

A bicycle wheel has diameter in. How far does the bike travel in one full rotation?

1. Circumference or area?
2. Set up the equation.
3. Give exact and approximate answers.

Answer: Circumference; in

Practice: Choose Formula and Solve

Label C or A first, then solve. (π ≈ 3.14)

1. Rim of a circular mirror, cm — trim needed?
2. Circular rug, ft — square feet of carpet?
3. Circular track curve, m — length of one arc?
4. Sprinkler circle, m — area watered?
5. Clock face, cm — area of the face?
6. Hot tub edge, ft — border tiling length?

Answers: Check Formula Selection Practice

When the Shape Is Not a Full Circle

Real problems often involve partial circles or combined shapes.

• Semicircle: half a circle —
• Composite figure: circle combined with a polygon
• Strategy: break into parts; apply formulas to each; add or subtract

Applied Problem 1: Fencing Cost

Problem: Pool with ft. Fencing costs $3/ft. Total cost?

Step 1: Circumference (fencing = boundary)

Step 2: Cost

Applied Problem 2: Semicircular Window

Problem: A semicircular window has m. Find the area.

Step 1: Radius: m

Step 2: Semicircle = of full circle

Semicircle area = half the full-circle area

Applied Problem 3: Composite Figure

Problem: Room 10 × 8 m; rug m. Uncovered floor area?

Practice: Four Applied Circle Problems

Label C or A first, show all steps. (π ≈ 3.14)

1. Sprinkler covers m. What area is watered?
2. Semicircular rug, ft. What is the area?
3. Pool ft. Safety rope goes halfway around. How long?
4. Square tile 20 × 20 cm, circle cm painted inside. Area NOT painted?

Answers: Check Your Applied Practice

1. m²
2. ; ft²
3. ft
4. cm²

Watch for: treating problem 3 as area (it's a distance — linear units); skipping the halve-the-diameter step in problem 2.

Summary: Key Takeaways and Warnings

• →
• Sector rearrangement →
• Going around → C · Covering inside → A
• Use radius not diameter in area formula
• Find first when given
• C: linear units · A: square units
• Exact: leave π · Approx: use 3.14

What's Next: Cylinders and Volume

In 7.G.B.6, you'll apply circle formulas to 3D solids:

• Surface area: (2 circles + a rectangle)
• Volume: (circle area × height)

drives both — every cylinder problem starts with circle area.